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## Abstract

The meridional and vertical eddy fluxes of sensible heat produced by small-amplitude growing baroclinic waves are calculated using solutions to the two-level model with horizontal shear in the mean flow. The results show that the fluxes are primarily dependent on the local baroclinicity, i.e., the local value of the isentropic slopes in the mean state. Where the slope exceeds the critical value, the transports are poleward and upward; where the slope is less than the critical value, the transports are equatorward and downward.

These results are used to improve an earlier parameterization of the tropospheric eddy fluxes of sensible heat based on Eady's model. Comparisons with observations show that the improved parameterization reproduces the observed magnitude and sign of the eddy fluxes and their vertical variations and seasonal changes, but the maximum in the poleward flux is too near the equator. The corresponding parameterizations for the eddy coefficients describing the transport of any conserved quantity are given.

## Abstract

The meridional and vertical eddy fluxes of sensible heat produced by small-amplitude growing baroclinic waves are calculated using solutions to the two-level model with horizontal shear in the mean flow. The results show that the fluxes are primarily dependent on the local baroclinicity, i.e., the local value of the isentropic slopes in the mean state. Where the slope exceeds the critical value, the transports are poleward and upward; where the slope is less than the critical value, the transports are equatorward and downward.

These results are used to improve an earlier parameterization of the tropospheric eddy fluxes of sensible heat based on Eady's model. Comparisons with observations show that the improved parameterization reproduces the observed magnitude and sign of the eddy fluxes and their vertical variations and seasonal changes, but the maximum in the poleward flux is too near the equator. The corresponding parameterizations for the eddy coefficients describing the transport of any conserved quantity are given.

## Abstract

Eady's (1949) model is used to study the non-geostorphic baroclinic stability problem. Growth rates for various types of perturbations are found as a function of the Richardson number, Ri The results indicate that the conventional baroclinic instabilities dominate if Ri > 0.95; symmetric instabilities dominate if 1/4 Ri > 0.95; and symmetric instabilities dominate if Ri < 1/4. It is suggested that symmetric instabilities may play an important role in the dynamics of the atmospheres of the major planets of the solar system.

## Abstract

Eady's (1949) model is used to study the non-geostorphic baroclinic stability problem. Growth rates for various types of perturbations are found as a function of the Richardson number, Ri The results indicate that the conventional baroclinic instabilities dominate if Ri > 0.95; symmetric instabilities dominate if 1/4 Ri > 0.95; and symmetric instabilities dominate if Ri < 1/4. It is suggested that symmetric instabilities may play an important role in the dynamics of the atmospheres of the major planets of the solar system.

## Abstract

The results of Parts I and II are used to calculate the transports of heat and momentum that accompany growing baroclinic instabilities in Eady's model. The transports are calculated for both the conventional (“geostrophic”) kind of baroclinic instability and for symmetric instability, without any restriction on the stratification, as measured by the Richardson number. The transports are calculated consistently to second order in the amplitude expansion of stability theory, so that the transports are the sum of an eddy transport term and a mean transport term.

The results show that both kinds of instability always transport heat upward and poleward, and always transport zonal momentum downward. Under geostrophic conditions the horizontal transport of zonal momentum depends on the horizontal shear of the basic flow. This shear is neglected in Eady's model so the horizontal momentum transports calculated here only contain the non-geostrophic contribution to the transport. The results show that this non-geostrophic transport is always equatorward for geostrophic instability, but for symmetric instability it may be either equatorward or poleward depending on the value of the Richardson number. It is suggested that the equatorward transport of zonal momentum by geostrophic instability is a more likely mechanism for Jupiter's equatorial acceleration than the transport by symmetric instability.

## Abstract

The results of Parts I and II are used to calculate the transports of heat and momentum that accompany growing baroclinic instabilities in Eady's model. The transports are calculated for both the conventional (“geostrophic”) kind of baroclinic instability and for symmetric instability, without any restriction on the stratification, as measured by the Richardson number. The transports are calculated consistently to second order in the amplitude expansion of stability theory, so that the transports are the sum of an eddy transport term and a mean transport term.

The results show that both kinds of instability always transport heat upward and poleward, and always transport zonal momentum downward. Under geostrophic conditions the horizontal transport of zonal momentum depends on the horizontal shear of the basic flow. This shear is neglected in Eady's model so the horizontal momentum transports calculated here only contain the non-geostrophic contribution to the transport. The results show that this non-geostrophic transport is always equatorward for geostrophic instability, but for symmetric instability it may be either equatorward or poleward depending on the value of the Richardson number. It is suggested that the equatorward transport of zonal momentum by geostrophic instability is a more likely mechanism for Jupiter's equatorial acceleration than the transport by symmetric instability.

## Abstract

No abstract available.

## Abstract

No abstract available.

## Abstract

Two-layer models of baroclinic instability predict that there is a critical temperature gradient separating conditions which are stable from those which are baroclinically unstable. In continuous models this critical gradient corresponds to a transition from conditions where the dominant baroclinic instabilities are inefficient at transporting heat to conditions where they are efficient. Zonal mean meridional temperature gradients in the atmosphere are compared with this critical gradient. For averages over periods longer than a few months the observed mid and mean tropospheric gradients never appreciably exceed the critical gradient. In fact they coincide remarkably closely with it in mid and high latitudes in all seasons in spite of strong seasonal changes in the forcing. This behavior shows that a very rapid transition must exist between conditions where eddy fluxes are inefficient to conditions where they are highly efficient. Thus, the primary effect of baroclinic eddies on the meridional temperature structure is to limit the gradients from becoming appreciably supercritical. This behavior allows one to take into account quite accurately the effect of the eddy fluxes on temperature structure without calculating the eddy fluxes explicitly, simply by adjusting the temperature gradients so that they never exceed the critical value. This baroclinic adjustment process is illustrated by incorporating it into a one-dimensional energy balance climate model. The results show that the process enhances the stability of the current climate to changes in the solar constant.

## Abstract

Two-layer models of baroclinic instability predict that there is a critical temperature gradient separating conditions which are stable from those which are baroclinically unstable. In continuous models this critical gradient corresponds to a transition from conditions where the dominant baroclinic instabilities are inefficient at transporting heat to conditions where they are efficient. Zonal mean meridional temperature gradients in the atmosphere are compared with this critical gradient. For averages over periods longer than a few months the observed mid and mean tropospheric gradients never appreciably exceed the critical gradient. In fact they coincide remarkably closely with it in mid and high latitudes in all seasons in spite of strong seasonal changes in the forcing. This behavior shows that a very rapid transition must exist between conditions where eddy fluxes are inefficient to conditions where they are highly efficient. Thus, the primary effect of baroclinic eddies on the meridional temperature structure is to limit the gradients from becoming appreciably supercritical. This behavior allows one to take into account quite accurately the effect of the eddy fluxes on temperature structure without calculating the eddy fluxes explicitly, simply by adjusting the temperature gradients so that they never exceed the critical value. This baroclinic adjustment process is illustrated by incorporating it into a one-dimensional energy balance climate model. The results show that the process enhances the stability of the current climate to changes in the solar constant.

## Abstract

Bergeron first suggested that atmospheric frontogenesis is caused by horizontal wind deformation fields acting on pre-existing horizontal temperature gradients. A three-dimensional time-dependent mathematical model of the atmosphere which incorporates this process through the initial conditions and boundary conditions is formulated. Dissipative processes are neglected and the equations are approximated by assuming that the Richardson number is initially large. The resulting equations are then solved analytically. The solution shows, under certain conditions, fronts developing with properties similar to many atmospheric fronts, thereby giving support to Bergeron's hypothesis.

## Abstract

Bergeron first suggested that atmospheric frontogenesis is caused by horizontal wind deformation fields acting on pre-existing horizontal temperature gradients. A three-dimensional time-dependent mathematical model of the atmosphere which incorporates this process through the initial conditions and boundary conditions is formulated. Dissipative processes are neglected and the equations are approximated by assuming that the Richardson number is initially large. The resulting equations are then solved analytically. The solution shows, under certain conditions, fronts developing with properties similar to many atmospheric fronts, thereby giving support to Bergeron's hypothesis.

## Abstract

The problem of the steady symmetric motion of a Boussinesq fluid is considered for a system with small aspect ratio. It is assumed that the motion is driven by applying a periodic heat flux to the horizontal boundaries. Solutions are first found for a non-rotating system in which nonlinear effects are small, but not zero. The solutions show that if the fluid is heated from above, the meridional circulation tends to be concentrated near the upper boundary at the point where the cooling is a maximum; when the fluid is heated from below the meridional circulation tends to be concentrated near the lower boundary at the paint where the heating is a maximum.

Then, it is shown for a non-rotating system that when nonlinear effects are dominant, vertical boundary layers must form. These vertical boundary layers form at points where the horizontal velocity is zero, and are characterized by small horizontal velocities and temperature gradients, but large vertical velocities and horizontal diffusion. By means of scaling analysis, the scales and magnitudes of the variables are determined for both the internal boundary layers and the boundary layers along the horizontal boundaries, when nonlinear effects are dominant.

Next, the effect of rotation is considered, and it shown that exactly the same sorts of vertical boundary layers will form in a rotating system. Scaling analysis is again used to show that in this case the horizontal boundary layers near the internal boundary layers are of the same kind as in the non-rotating case, but far enough away from the internal boundary layers they merge into a nonlinear Ekman layer.

Finally, some possible geophysical applications are considered. The model of the atmospheric circulations on Venus proposed by Goody and Robinson is found to agree qualitatively with the results presented here, but the quantitative results for the internal boundary layer, or mixing region, are found to differ considerably. Also, estimates are made for the internal boundary layer which would accompany a Hadley cell similar to that found in the earth's tropical region. It is found that the rising motions will occur over a region about 200 km in width. This result suggests that the nonlinear process which produces these internal boundary layers may be one of the important processes in determining the structure of the Intertropical Convergence Zone. Finally, the identification of the narrow sinking regions as another example of the kind of internal boundary layer studied here is considered, but in this case the magnitudes and scales are not plausible.

## Abstract

The problem of the steady symmetric motion of a Boussinesq fluid is considered for a system with small aspect ratio. It is assumed that the motion is driven by applying a periodic heat flux to the horizontal boundaries. Solutions are first found for a non-rotating system in which nonlinear effects are small, but not zero. The solutions show that if the fluid is heated from above, the meridional circulation tends to be concentrated near the upper boundary at the point where the cooling is a maximum; when the fluid is heated from below the meridional circulation tends to be concentrated near the lower boundary at the paint where the heating is a maximum.

Then, it is shown for a non-rotating system that when nonlinear effects are dominant, vertical boundary layers must form. These vertical boundary layers form at points where the horizontal velocity is zero, and are characterized by small horizontal velocities and temperature gradients, but large vertical velocities and horizontal diffusion. By means of scaling analysis, the scales and magnitudes of the variables are determined for both the internal boundary layers and the boundary layers along the horizontal boundaries, when nonlinear effects are dominant.

Next, the effect of rotation is considered, and it shown that exactly the same sorts of vertical boundary layers will form in a rotating system. Scaling analysis is again used to show that in this case the horizontal boundary layers near the internal boundary layers are of the same kind as in the non-rotating case, but far enough away from the internal boundary layers they merge into a nonlinear Ekman layer.

Finally, some possible geophysical applications are considered. The model of the atmospheric circulations on Venus proposed by Goody and Robinson is found to agree qualitatively with the results presented here, but the quantitative results for the internal boundary layer, or mixing region, are found to differ considerably. Also, estimates are made for the internal boundary layer which would accompany a Hadley cell similar to that found in the earth's tropical region. It is found that the rising motions will occur over a region about 200 km in width. This result suggests that the nonlinear process which produces these internal boundary layers may be one of the important processes in determining the structure of the Intertropical Convergence Zone. Finally, the identification of the narrow sinking regions as another example of the kind of internal boundary layer studied here is considered, but in this case the magnitudes and scales are not plausible.

## Abstract

No abstract available.

## Abstract

No abstract available.

## Abstract

A simple model for the structure of a non-rotating Hadley regime in an atmosphere with large thermal inertia is developed. The radiative fluxes are estimated by using a linearization about the radiative equilibrium state and the dynamical fluxes are estimated by using scaling analysis. The requirement that differential heating by these fluxes be in balance in both the meridional and vertical directions leads to two equations for the mean static stability and meridional temperature contrast. The solution depends on two parameters: the strength of the radiative heating, as measured by the static stability *A _{e}
* of the radiative equilibrium state; and the ratio of the time it takes an external gravity wave to traverse the atmosphere to the time it would take the atmosphere to cool off radiatively, denoted by ε.

In the deep Venus atmosphere ε ≈ 10^{−5}; the equations are therefore analyzed in the limit ε → 0. The large-scale dynamics has virtually the same effect on the lapse rate as small-scale convection: if *A _{e}
* > 0 the radiative lapse rate is unchanged, while if

*A*< 0 the lapse rate becomes subadiabatic, but only by an amount of order ε

_{e}^{⅔}. Therefore, one need not invoke convection to explain the approximate adiabatic lapse rate in the Venus atmosphere, but a greenhouse effect is necessary to explain the high surface temperatures. The other properties of the solutions when

*A*< 0 are consistent with observational evidence for the deep atmosphere: the horizontal velocities are typically ∼2 m sec

_{e}^{−1}, the vertical velocities ∼½ cm sec

^{−1}, and the meridional temperature contrast is unlikely to exceed 0.1K.

The same approach is used to study the time-dependent problem and determine how long it would take for a perturbed atmosphere to reach equilibrium. If *A _{e}
* > 0 the adjustment is primarily governed by the radiative time scale, which is about 100 earth years for the deep Venus atmosphere. If

*A*< 0 the adjustment is governed by an advective time scale which may be as short as 20 earth days. Published numerical studies of the deep circulation have only treated the first case, but their integrations were not carried beyond about 200 earth days and therefore do not describe true equilibrium states. Only the second case,

_{e}*A*< 0, is consistent with the observations and it would be relatively easy to study numerically.

_{e}## Abstract

A simple model for the structure of a non-rotating Hadley regime in an atmosphere with large thermal inertia is developed. The radiative fluxes are estimated by using a linearization about the radiative equilibrium state and the dynamical fluxes are estimated by using scaling analysis. The requirement that differential heating by these fluxes be in balance in both the meridional and vertical directions leads to two equations for the mean static stability and meridional temperature contrast. The solution depends on two parameters: the strength of the radiative heating, as measured by the static stability *A _{e}
* of the radiative equilibrium state; and the ratio of the time it takes an external gravity wave to traverse the atmosphere to the time it would take the atmosphere to cool off radiatively, denoted by ε.

In the deep Venus atmosphere ε ≈ 10^{−5}; the equations are therefore analyzed in the limit ε → 0. The large-scale dynamics has virtually the same effect on the lapse rate as small-scale convection: if *A _{e}
* > 0 the radiative lapse rate is unchanged, while if

*A*< 0 the lapse rate becomes subadiabatic, but only by an amount of order ε

_{e}^{⅔}. Therefore, one need not invoke convection to explain the approximate adiabatic lapse rate in the Venus atmosphere, but a greenhouse effect is necessary to explain the high surface temperatures. The other properties of the solutions when

*A*< 0 are consistent with observational evidence for the deep atmosphere: the horizontal velocities are typically ∼2 m sec

_{e}^{−1}, the vertical velocities ∼½ cm sec

^{−1}, and the meridional temperature contrast is unlikely to exceed 0.1K.

The same approach is used to study the time-dependent problem and determine how long it would take for a perturbed atmosphere to reach equilibrium. If *A _{e}
* > 0 the adjustment is primarily governed by the radiative time scale, which is about 100 earth years for the deep Venus atmosphere. If

*A*< 0 the adjustment is governed by an advective time scale which may be as short as 20 earth days. Published numerical studies of the deep circulation have only treated the first case, but their integrations were not carried beyond about 200 earth days and therefore do not describe true equilibrium states. Only the second case,

_{e}*A*< 0, is consistent with the observations and it would be relatively easy to study numerically.

_{e}## Abstract

It is suggested that the apparent lag of Jupiter's mean rotation rate in extratropical latitudes (System II) behind the rotation rate of Jupiter's radio emissions (System III) is caused by the difference between phase speeds and true speeds in extratropical latitudes. An estimate of the difference based on the formula for the phase speed of Rossby waves agrees with the difference calculated from the two rotation rates.

## Abstract

It is suggested that the apparent lag of Jupiter's mean rotation rate in extratropical latitudes (System II) behind the rotation rate of Jupiter's radio emissions (System III) is caused by the difference between phase speeds and true speeds in extratropical latitudes. An estimate of the difference based on the formula for the phase speed of Rossby waves agrees with the difference calculated from the two rotation rates.